This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306670 #17 Mar 05 2019 08:55:35 %S A306670 345,465,468,1332,1545,1833,1872,2628,2737,2769,3105,3145,3585,3657, %T A306670 3945,4081,4100,4185,4212,4345,5328,6465,6516,6785,6945,7105,7488, %U A306670 8428,8569,8625,8961,10257,10512,10785,10833,10945,11625,11988,12132,12865 %N A306670 Numbers k with exactly three distinct prime factors and such that cototient(k) is a square. %C A306670 The integers with only one prime factor and whose cototient is a square are in A246551. The integers with two prime factors and whose cototient is a square are in A323916, and the subsequences A323917 and A323918. %C A306670 There are exactly three different families of integers which realize a partition of this sequence. See the file "Subfamilies and subsequences" (& III) in A063752 for more details, proofs with data, comments, formulas and examples. %F A306670 1st family: The primitive terms are p*q*r with p,q,r primes and cototient(p*q*r) = p*q*r-(p-1)*(q-1)*(r-1) = M^2. These primitives generate the entire family formed by the numbers k = p^(2s+1) * q^(2t+1) * r^(2u+1) with s,t,u >=0, and cototient(k) = (p^s * q^t * r^u * M)^2. %F A306670 2nd family: The primitive terms are p^2 *q * r with p,q,r primes and cototient(p^2 * q * r) = p * (p*q*r-(p-1)*(q-1)*(r-1)) = M^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t+1) * r^(2u+1) with s>=1, t,u >=0, and cototient(k) = (p^(s-1) * q^t * r^u * M)^2. %F A306670 3rd family: The primitive terms are p^2 * q^2 * r with p,q,r primes and cototient(p^2 * q^2 * r) = p * q * (p*q*r-(p-1)*(q-1)*(r-1)) = M^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t) * r^(2u+1) with s,t>=1, u >=0, and cototient(k) = (p^(s-1) * q^(t-1) * r^u * M)^2. %e A306670 1st family: 2769 = 3 * 13 * 71 and cototient(2769) = 33^2. %e A306670 2nd family: 14841 = 3^2 * 17 * 97 and cototient(14841) = 75^2. %e A306670 3rd family: 1872 = 2^4 * 3^2 * 13 and cototient(1872) = 36^2. %Y A306670 Subsequence of A063752. %Y A306670 Cf. A246551 (only one prime factor), A323916, A323917, A323918 (two prime factors), A000396 (even perfect numbers). %K A306670 nonn %O A306670 1,1 %A A306670 _Bernard Schott_, Mar 04 2019